Drawing on Desargues

Drawing
on Desargues
ANNALISA CRANNELL
AND
STEPHANIE DOUGLAS
D E S A R G U E S ’ S T R I A N G L E T H E O R E M : Two triangles
that are perspective from a point are also perspective from a
line.
bout four years before Girard Desargues wrote the
Leçon de Te´ne`bres containing the preceding theorem—
a theorem that bears his name to this day—he wrote a
12-page treatise on artistic perspective [6]. Desargues’s
perspective treatise was a bit of an IKEA manual: it describes
how to draw a gazebo-like object of certain proportions
without resorting to the use of vanishing points (see
Figure 1), and Desargues did not seem to expect that lay
artists would apply its techniques to more general objects.
The book was an explanation of an example, not an
explanation of a theory; Desargues’s art treatise does not
contain Desargues’s triangle theorem. Still, projective geometry has flirted with perspective art ever since.
Desargues’s more mathematical Leçon de Te´ne`bres is lost;
the closest surviving relative comes in the final several pages
of a book by one of Desargues’s acolytes, Bosse [2], published
a dozen years later, pulls together both the artistic and the
mathematical work of Desargues. But Bosse proof of Desargues’s triangle theorem is more mathematical than artistic.
It uses the cross-ratio, a tool found in very few artists’ repertoires. And the accompanying diagram? If you, like most
people, have a hard time deciphering Bosse’s arrangement
(Figure 2), you can try puzzling it out by noting that triangles
abl and DEK are perspective from both the point H and the
line cfg. Or you could realize that it’s really not a very helpful
diagram.
Bosse’s is the first in a centuries-long tradition of Desargues’s Theorem illustrations, few of which require as much
effort on the part of the reader as Bosse’s. Still, if you search
through the projective geometry books in your library, you
are likely to find a lot of pencil marks. In our own searches
through some 20 different volumes (see [1], [3]–[5], [7]–[10]
and [13]–[29]), we found we had to do a lot of erasing if we
wanted a ‘‘clean’’ version of the author’s Desargues’s triangle
figures. People who have used these books over the years
have shaded in triangles, darkened lines, and circled important points so that they could read the diagram more easily.
A
The point of this article is to illustrate—literally—Desargues’s theorem. Many authors have shown that it is possible
to ‘‘lift’’ a planar Desargues configuration into 3-dimensions;
we will instead view a planar configuration as a drawing of
objects already in 3 dimensions. Doing so gives us a lovely
and paradoxical academic symmetry: perspective art uses
geometrical tools to portray a 3-d world on a 2-d canvas
whereas we give 2-d geometrical drawings a 3-d perspective
interpretation.
Definitions
Figure 3 shows two common physical ways we might project
objects from our 3-d world onto a canvas: the candle flame
projects a shadow of the rabbit onto the floor and wall, and
the pinhole projects an image of the thinker onto the wall. In
projective geometry we formalize these physical notions by
saying that two triangles X ; Y R3 are perspective from a
point O if there is a one-to-one correspondence between the
vertices of these triangles so that whenever the vertex VX 2 X
corresponds to the vertex VY 2 Y; the three points VX ; VY ;
and O are collinear. We call O the center of the projection.
We’ll have occasion later to refer to ‘‘shadow’’ perspectivities
(for which a triangle and its image are on the same side of the
center) and ‘‘pinhole’’ perspectivities (for which the triangle
and its image are on opposite sides of the center). As Figure 4
shows, these are not the only kinds of perspectivities.
We can say that the triangles X and Y are perspective from
a line ‘ if there is a one-to-one correspondence between the
three lines (the extended edges) of these triangles so that
whenever the line ‘X X corresponds to the line ‘Y Y; the
lines ‘X ; ‘Y ; and ‘ meet at a common point. In other words,
the three points where pairs of corresponding lines intersect
all lie on ‘. We call the line ‘ the axis. In Figure 5, the lines of
the shaded triangle on the paper and their reflections in the
mirror are perspective from the line where the mirror meets
the paper.
It is not true in general that two objects that are perspective
from a point are also perspective from a line (indeed, the
difference between these two kinds of perspectivities forms
the basis for some spiffy optical illusions; think of M. C.
Escher’s Waterfall or the sculpture of the Penrose Triangle in
Ó 2012 Springer Science+Business Media, LLC, Volume 34, Number 2, 2012
7
DOI 10.1007/s00283-011-9271-y
Figure 1. The plate illustrating Desargues’s Perspective.
East Perth, Australia). Desargues’s theorem and its converse
tell us, however, that these two notions are equivalent for
triangles. An intuitive and rather breezy explanation of this
phenomenon is that a triangle (unlike a rabbit, a hand, or a
thinker) uniquely defines a plane. Two triangles perspective
from a point can be interpreted as a 2-d drawing of triangles in
3-space, lying in two planes whose intersection (as in the
example of the paper and mirror of Figure 5) forms the axis.
We will not prove Desargues’s theorem, but we will use the
above breezy interpretation to draw our own pictures of the
theorem.
Drawing on Coxeter
There are some remarkable commonalities among the published diagrams illustrating Desargues’s Theorem. Figure 6
Figure 2. Desargues’s theorem as illustrated by Bosse, from
[11]. We have enlarged the labels from the original diagram.
gives three fairly typical examples. With one exception, all the
diagrams we have seen illustrate a shadow-type projection;
Horadam [16] bucks the trend by showing a pinhole-type
projection. More significantly, in every one of the books in the
collection we examined, the diagrams illustrate a case where
the perspectivity maps the filled-in, finite interior of one triangle onto the filled-in, finite interior of the other triangle.
That is, there are no diagrams that look like Figure 4. As we
will see, this interior-to-interior projection is often incorrect.
For all else that follows, we will interpret and reinterpret
(indeed, literally draw on) the diagram that appears in what is
arguably the best-known book of its kind, Coxeter’s Projective Geometry [4]. We use this diagram not only because of its
AUTHORS
.........................................................................................................................................................
ANNALISA CRANNELL received her B.A.
STEPHANIE DOUGLAS is a 2012 graduate of
from Bryn Mawr College and her Ph.D.
from Brown University. Her mathematical
interests have morphed over the years
from nonlinear PDEs to discrete dynamical
systems to – most recently – the applications of projective geometry to perspective
art. Together with Marc Frantz, she is a
coauthor of the recent Viewpoints: Mathematical Perspective and Fractal Geometry
in Art.
Franklin & Marshall College, majoring in
astrophysics. She enjoys dabbling in mathematics on the occasions when it involves art.
She intends to study astrophysics in graduate
school.
Department of Mathematics
Franklin & Marshall College
Box 3003
Lancaster, PA 17604-3003
USA
e-mail: [email protected]
8
THE MATHEMATICAL INTELLIGENCER
Department of Mathematics
Franklin & Marshall College
Box 3003
Lancaster, PA 17604-3003
USA
e-mail: [email protected]
Figure 3. A shadow projection and a pinhole projection.
Figure 4. We project a vertical triangle onto a horizontal
plane. The parts of the triangle above the center of projection
get sent via a pinhole projection to an infinite triangle; the
parts below the triangle get sent via a shadow projection to an
infinite trapezoid. The dividing line segment gets sent ‘‘to a
line at infinity.’’.
Figure 5. The lines on the paper and their images in the
mirror are perspective from a line (the line where the mirror
meets the paper).
popularity, but also because Coxeter’s drawing of Desargues’s
Theorem is simple in the extreme: it shows no shading or
variation in line-weight, only ten line segments and ten
points that are indistinguishable in style. Figure 7 shows the
configuration, although we altered the diagram slightly by
removing Coxeter’s labels and giving six of the ten points in
his diagram our own new labels. The explanation of these
new labels will become apparent later.
Coxeter chose as the center of his perspectivity the point
we call A3; if we shade in the resulting triangles that follow
from this choice, we get a figure like that of Eves (Figure 6). If
we add a few details, we get Figure 8: an illustration of a light
shining through an object on a window, creating a shadow. In
addition to ‘‘embedding’’ each triangle in a plane, we have
also used standard perspective techniques (such as overlapping and variable line weights) to imply that some lines
are closer to the viewer of the diagram than others. In this
image, the conclusion that each edge of the triangle must
meet its shadow at the axis becomes visually obvious.
Let’s step back to consider why the readers of our copy of
Coxeter made so many additional pencil marks in the book.
All ten line segments in the diagram look the same, and that’s
because, in an important geometrical sense, they all are the
same. Coxeter’s Desargues configuration has the property
that each of the ten line segments contains three points; each
of the ten points lies on three lines. So what happens if we
choose a different point to be the center? The reader can
verify easily that choosing a point as center automatically
determines the triangles and the axis. In what follows, these
choices and their implications become the variations on our
theme.
The next center we chose is the one we named B3. Again,
we shade the interiors of the resulting triangles. In this case,
because the axis passes through the interior of the triangles,
we use standard overlapping techniques to indicate that
some parts of the triangles are close and others are far. Does
Figure 9(a) show a triangle passing through its own shadow?
We think this diagram is visually understandable and
believable if we reinterpret it slightly to say that each one of
these triangles casts the same shadow on a given plane
(Figure 9(b)), in the same way that the rabbit of Figure 3 casts
the same shadow as a hand would. [An exercise for the
reader: this coloring of Coxeter’s diagram gives us the same
configuration as Bosse’s diagram. Can you see that for
yourself?]
But our variations on this theme get weirder at other
points. Both points A3 and B3 have the property that they are
endpoints of their three line segments—the subscript ‘‘3’’ in
the name of each point stands for the number of line segments ending at that point. In Figure 10, we choose other
points as center and then color the interiors of the triangles as
we did before. Our diagrams look a bit like the earlier ones,
but a closer look tells us something interesting is going on.
Now what does the axis mean? If C2 is the center, why does the
axis pass through one triangle but not through the other?
Similarly, what is going on in the diagram whose center is D2?
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Figure 6. Desargues’s theorem, as illustrated by Eves [9], Veblen and Young [28], and Horadam [16].
Figure 7. Coxeter’s diagram of Desargues’s theorem, with
Crannell’s labels. This Desargues’s configuration has the
property that each of the ten line segments contains three
points; each of the ten points lies on three lines.
Or in the diagram whose center is E1? The center of the projection F0 is contained in the interior of three segments. That
is, F0 is like the hole of a pinhole camera, . . . but how do we
understand the dark line of projectivity? Why is it where it is,
and not (for example) vertical, between the two triangles as it
is in Horadam’s diagram (Figure 6)?
There are two reasons for the visual confusion of the
diagrams in Figures 10. The first reason is that our own
location in space (our perspective, so to speak) matters.
Let us illustrate this first concern in the case of the last
example, for which the center is F0. When two planes
intersect, they divide three-space into four quadrants. When
we imagine a pinhole projection, we most easily imagine
ourselves in the same quadrant as the pinhole, as Figure 11(a). In this figure, the axis appears to lie between the
two triangles, which matches our intuition. But when we
rotate the diagram in Figure 11(a), we get Figure 11(b) . . .
and this latter view matches Coxeter’s diagram. That is, using
artistic techniques to create a sense of appropriate 3-d space
allows us to portray the diagram in a way that seems ‘‘correct.’’
The second reason for the visual confusion is more substantive. It has to do with the parts of the figures we chose to
shade. When two triangles are perspective from a center
O, although the vertices of one triangle project from O on to
the vertices of the other, the interior of a triangle in a Desargues’s diagram does not always project onto the interior of the
other triangle, for reasons that Figure 12 illustrates. The disconnected projection of line segments in Figure 12 is related
to the disconnected projection of triangles from Figure 4. As
we noted previously, this case was not addressed in any of the
books that we examined. This is why Desargues’s theorem
properly speaks of lines, not of line segments—a distinction
that does not seem to be apparent in most graphic depictions
of the theorem.
Let us give an example of how this ‘‘inside-out’’ projection
applies to Coxeter’s figure, looking at the projection centered
at O = D2. Figure 13 relabels the points in a way that will help
us describe overlappings.
The line segment A0 OA never crosses the dark axis ‘. That
tells us that the line segment A0 A lies in the same quadrant as
the center O. The segment OBB0 tells us that B is on a halfplane near the center O whereas B0 is on a far one (the
Figure 8. Coxeter’s diagram with the center of projectivity at A3, oriented (left) as in Coxeter and (right) with the axis represented
in a horizontal position. The axis ‘ is represented by the intersection of the two planes containing the respective triangles.
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THE MATHEMATICAL INTELLIGENCER
(a)
(b)
Figure 9. (a) Coxeter’s diagram with the center of projectivity at B3. (b) We may think of this diagram as saying that many
different triangles can cast the same shadow.
Figure 10. Coxeter’s diagram with the center of projectivity at C2, D2, E1, and F0. We fill in the interiors of the resulting triangles,
but something strange is going on here. How should we interpret these diagrams?.
(a)
(b)
Figure 11. A pinhole projection. In (a), we are in the same quadrant with the pinhole. In (b), we rotate the preceding pinhole
figure to get an interpretation of Coxeter’s diagram centered at F0.
intersection of the axis with the segment BC 0 confirms this).
Likewise, C is on a near half-plane and C 0 is on a far one. That
is, both triangles cross the axis. See Figure 14.
The last step in drawing an interpretation of the diagram is
to note that the interiors of the triangles in this diagram aren’t
really projective images of each other, even though the lines
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Figure 12. The center O projects the points A and B on the
line ‘ to the points A0 and B0 on the line ‘0 , but it projects the
line segment AB to the complement of the line segment A0 B0 .
Figure 15. A recoloring of Figure 14 shows us that the
interior of the finite triangle projects to the union of an infinite
trapezoid and an infinite triangle. Where does the break
happen? The plane passing through O and parallel to the
image plane intersects the finite triangle in a line that gets sent
to infinity. (Compare with Figures 16 and 4.).
Figure 13. Coxeter’s diagram with the center of projectivity at
D2. We have labeled the points and the axis so that we can try
to figure out where in the world things are.
Figure 16. If we rotate Figure 15, we see something that
reminds us of Figure 4, as seen from above. In this figure,
though, the object triangle begins to dip below the image
plane.
Figure 14. A projective image of a strange-looking projectivity. The vertices of the triangles are projective images of each
other, but the interiors of the triangles do not map onto one
another. The shading is therefore wrong.
that form the edges of the triangles are. Instead, as with
Figure 4, the projective image of the finite triangle is a combination of an infinite trapezoid (toward the top of Figure 15)
and an infinite triangle (on the bottom right side of the figure).
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THE MATHEMATICAL INTELLIGENCER
The trapezoid is the result of a shadow projection; the triangle
is the result of a pinhole projection. If we rotate Figure 15
clockwise 90 degrees, as in Figure 16, the similarity to Figure 4 is all the more striking.
We may proceed in the same way to draw perspective
views of Coxeter’s diagram with the center at C2 or E1, as in
Figures 17 and 18. For example, we might see the arrangement whose center is at E1 as a triangle passing through its
image plane (as in the left of Figure 18); if we rotate this
configuration toward us, we will get the Coxeter diagram on
the right.
Given a projective diagram (such as Coxeter’s), it is a hardbut-worthwhile exercise to figure out how to represent that
diagram accurately as a meta-projection. Readers who want
to see how tricky this is should cover up the figures in this
article and try their hands at C2 or E1! And often, as the pinhole
example shows, it’s helpful to have several meta-versions,
(b)
(a)
Figure 17. The configuration for Coxeter’s diagram with the center at C2, two ways. In (a), we view the configuration as seen
from one side, with an eye indicating the viewing location for the image in (b), which matches Coxeter.
Figure 18. The configuration for Coxeter’s diagram with the center at E1, two ways: (left) as seen from one side; (right) rotated
toward us and as seen from above, matching Coxeter’s figure. The points P, Q, and R are labeled to assist with understanding the
rotation.
Figure 19. If we see the coordinate axes configured as they are on the left, we are in the main quadrant (where x, y, and z are all
positive). Where are we if we see the axes as in the middle or rightmost configurations? This is a question Pat Oakley asked her
students.
some of which show the projectivity from a more intuitive
point of view.
A Matter of Viewpoint
As we have seen in the preceding section, it is a hard but
worthwhile exercise to interpret Desargues’s diagrams as
drawings of two triangles in space, each in a separate plane,
in which rays from a point O project vertices of one triangle
onto the other. Readers who want to see how tricky this is
should cover up Figures 17 and 18 and try their hands at the
diagrams with centers at C2 or E1! And often, as the pinhole
example shows, it’s helpful to have several meta-versions,
some of which show the projectivity from a more intuitive
point of view. Finding a ‘‘good’’ viewpoint, from which the
diagram can easily be interpreted, makes all the difference.
For this paper, the authors made several (sometimes
many) pencil-and-paper sketches of each diagram in trying to
‘‘see’’ the figure. When we got a representation that pleased
Ó 2012 Springer Science+Business Media, LLC, Volume 34, Number 2, 2012
13
us, we created a more formal version using drawing software
(to be specific, Lineform, commercially available from Freeverse.) In this way, the process reminds us more of art than of
mathematics.
But of course, the mathematics drives these figures. Our
approach to Desargues’s diagrams was motivated in part by
an exercise that Pat Oakley of Goshen College cooked up for
her Calculus 3 students in response to the Viewpoints math/
art materials [12]. She had her students figure out where in R3
they were if they saw the axes in various configurations (see
Figure 19). It’s a great exercise!
[12] Frantz, M. and Crannell, A., Viewpoints: Mathematical Perspective and Fractal Geometry in Art, Princeton University Press,
Princeton (2011).
[13] Garner, L. E., An Outline of Projective Geometry, North Holland,
New York (1981).
[14] Hilbert, D. and Cohn-Vossen, S., Geometry and the Imagination,
Chelsea Publishing Company, New York (1952).
[15] Holgate, T. F., Projective Pure Geometry, MacMillan, New York
(1930).
[16] Horadam, A. F., A Guide to Undergraduate Projective Geometry,
Pergamon Press, Australia (1970).
[17] Levy, H., Projective and Related Geometries, Macmillan, New
York (1961).
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THE MATHEMATICAL INTELLIGENCER
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